The cross product of two sets A and B, denoted by A × B , eh times b comma  is the set of all ordered pairs with the first element in A and with the second element in B. In set-builder notation, you write:

A × B = { ( a , b ) |   a  is an element of  A , b  is an element of  B } eh times b equals left brace open eh comma b close vertical line eh . isanelementof eh comma b . isanelementof b right brace

For example, suppose A = { 1 , 2 } eh equals left brace 1 comma 2 right brace  and B = { 7 , 10 , 12 } . b equals left brace 7 comma 10 comma 12 right brace .  Then:

A × B = { ( 1 , 7 ) , ( 1 , 10 ) , ( 1 , 12 ) , ( 2 , 7 ) , ( 2 , 10 ) , ( 2 , 12 ) } eh times b equals left brace open 1 comma 7 close comma open 1 comma 10 close comma open 1 comma 12 close comma open 2 comma 7 close comma open 2 comma 10 close comma open 2 comma 12 close right brace

Given sets A and B, find A × B . eh times b .

40. A = { 1 , 2 , 3 } , B = { − 3 , − 2 , − 1 , 0 } eh equals left brace 1 comma 2 comma 3 right brace comma b equals left brace negative 3 comma negative 2 comma negative 1 comma 0 right brace
41. A = { π , 2 π , 3 π , 4 π } , B = { 2 , 4 } eh equals left brace pi comma 2 pi comma 3 pi comma 4 pi right brace comma b equals left brace 2 comma 4 right brace
42. A = { grape , apple , orange } , B = { jam , juice } eh equals left brace , grape , comma , apple , comma , orange , right brace comma b equals left brace jam comma , juice , right brace
43. A = { reduce , reuse , recycle } , B = { plastic } eh equals left brace , reduce , comma , reuse , comma , recycle , right brace comma b equals left brace , plastic , right brace

C Challenge

44. Use a Venn diagram to determine whether the statement ( A ∩ B ) ′ = A ′ ∩ B ′ open eh intersection b close prime equals eh prime intersection b prime  is true or false.
45. Reasoning Is the statement ( A ∪ B ) ∩ C = A ∪ ( B ∩ C ) open eh union b close intersection c equals eh union open b intersection c close  always, sometimes, or never true? Justify your answer.

Standardized Test Prep

SAT/ACT

46. Set X = { x | x  is a factor of  12 } x equals left brace x vertical line x . isafactorof . 12 right brace  and set Y = { y | y  is a factor of  16 } . y equals left brace y vertical line y . isafactorof . 16 right brace .  Which set represents X ∩ Y ? x intersection y question mark
    1. ø o with stroke
    2. { 1 , 2 , 4 } the set 1 comma 2 comma 4 end set
    3. { 0 , 1 , 2 , 4 } the set 0 comma 1 comma 2 comma 4 end set
    4. { 1 , 2 , 3 , 4 , 6 , 8 , 12 , 16 } the set 1 comma 2 comma 3 comma 4 comma 6 comma 8 comma 12 comma 16 end set
47. Which compound inequality is equivalent to | x + 4 | < 8 ? vertical line x plus 4 vertical line less than 8 question mark
    6. − 12 < x < 4 negative 12 less than x less than 4
    7. x < − 12  or  x > 4 x less than negative 12 , or , x greater than 4
    8. − 12 > x > 4 negative 12 greater than x greater than 4
    9. x > − 12  or  x < 4 x greater than negative 12 , or , x less than 4

Short Response

48. Suppose you earn $80 per week at your summer job. Your employer offers you a $20 raise or a 20% raise. Which should you take? Explain.

Mixed Review

See Lesson 3-7.

Solve each equation or inequality.

49. | x | = 4
50. | n | + 7 = 9
51. 4 | f − 5 | = 12 4 vertical line f minus 5 vertical line equals 12
52. 3| 3y + 2 | = 18
53. | 4 d | ≤ 20 vertical line 4 d vertical line less than or equal to 20
54. | x − 3 | ≥ 7 vertical line x minus 3 vertical line greater than or equal to 7
55. | 2 w + 6 | > 24 vertical line 2 w plus 6 vertical line greater than 24
56. 2| 3x | + 1 = 9

See Lesson 1-9.

Tell whether the ordered pair is a solution to the given equation.

57. x + 3 = y; (1, 4)
58. 2 x − 5 = y ; ( − 1 , 8 ) 2 x minus 5 equals y semicolon open negative 1 comma 8 close
59. 1 2 x + 7 = y ; ( 8 , 11 ) 1 half , x plus 7 equals y semicolon open 8 comma 11 close

Get Ready! To prepare for Lesson 4-1, do Exercises 60–63.

See Review p. 60.

Graph each point on the same coordinate grid.

60. (1, 4)
61. ( − 1 , − 5 ) open negative 1 comma negative 5 close
62. ( 3 , − 6 ) open 3 comma negative 6 close
63. ( − 2 , 1 ) open negative 2 comma 1 close

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